Zoology in the H\'enon family: twin babies and Milnor's swallows
Pierre Berger

TL;DR
This paper investigates the parameter space of Hénon-like maps, establishing a conjugacy to quadratic-like maps, proving a diffeomorphism between parameters and quadratic map coefficients, and demonstrating the existence of maps with exactly two attracting cycles.
Contribution
It introduces a new renormalization framework for Hénon-like maps, proving a diffeomorphism between parameters and quadratic map coefficients, and constructs maps with exactly two attracting cycles.
Findings
Existence of an open set of parameters conjugate to quadratic-like maps.
The parameter-to-coefficient map is a $C^d$-diffeomorphism.
Existence of Hénon maps with exactly two attracting cycles.
Abstract
We study -H\'enon-like families with two parameters . We show the existence of an open set of parameters , so that a renormalization chart conjugates an iterate of to a perturbation of . We prove that the map is a -diffeomorphism; as first numerically conjectured by Milnor in 1992. Furthermore, we show the existence of an open set of parameters so that displays exactly two different renormalized H\'enon-like maps whose basins union attracts Lebesgue a.e. point with bounded forward orbit. A great freedom in the choice of the renormalized parameters enables us to deduce in particular the existence of a (unperturbed) H\'enon map with exactly attracting cycles (an answer to a Question by Lyubich). The…
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Taxonomy
TopicsMathematical Dynamics and Fractals · Quantum chaos and dynamical systems · Advanced Differential Equations and Dynamical Systems
